#6749761

order	512
nilpotency class	3
isoclinism	isoclinic class of #6667900
automorphism	?
homology	Abelianization (H₁) = C₂⁵ · Schur multiplier (H₂) = C₂⁷

Aliases

No aliases for this group.

Subgroups

	subgroup	G/H
center, socle	C₂³	C₂³×D₄
commutator, Frattini, cosocle	C₂²×C₄	C₂⁵
prime radical	#2414	C₂²
LCS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁
UCS	C₁ ◁ C₂³ ◁ #2290 ◁ G
LECS	G ▷ C₂²×C₄ ▷ C₂ ▷ C₁

1395 conjugacy classes of subgroups.

Full subgroup data →

Presentation

minimal degree	μ(G) = 56
transitive degree	μ_t(G) = 256

A minimal faithful permutation representation — 5 generators on 56 points:

(1,2,4,6)(3,5,7,8)(9,10,14,17)(11,18,20,15)(12,21,13,16)(19,23,22,24)(25,26,29,31)(27,35,36,48)(28,39,40,33)(30,43,42,34)(32,44,45,38)(37,51,50,49)(41,47,55,54)(46,52,53,56);(1,3)(2,5)(4,7)(6,8)(9,11,14,20)(10,15,17,18)(12,19,13,22)(16,23,21,24)(26,31)(27,36)(33,39)(34,43)(37,50)(38,44)(47,54)(52,56);(3,7)(5,8)(25,27,29,36)(26,32,31,45)(28,37,40,50)(30,44,42,38)(33,46,39,53)(34,48,43,35)(41,56,55,52)(47,51,54,49);(9,12,14,13)(10,16,17,21)(11,19,20,22)(15,23,18,24)(25,28,29,40)(26,33,31,39)(27,37,36,50)(30,41,42,55)(32,46,45,53)(34,47,43,54)(35,49,48,51)(38,52,44,56);(3,7)(5,8)(9,13,14,12)(10,16,17,21)(11,19,20,22)(15,24,18,23)(28,40)(33,39)(37,50)(41,55)(46,53)(47,54)(49,51)(52,56)

Polycyclic presentation:

2^9|a^2=fhi,b^2=fi,c^2=i,d^2=f,e^2=fi,g^2=i,b^a=bf,c^a=cg,d^a=df,g^a=gi,c^b=ch,e^b=efh,g^c=gi,e^d=ei

Conjugacy classes & character table

71 conjugacy classes — view the full character table.

GAP

Construct this group in GAP via the SmallGroups library:

gap> SmallGroup(512, 6656957);